Quantum Field Theory of Many-Body Systems

Lecture (MVSpec)

Thomas Gasenzer

Tuesday, 11:15-13:00; Thursday, 11:15-13:00; INF 227 (KIP), SR 1.404. [LSF]

Exercises
Tutor: Markus Karl

Register and view group list here.
Classes take place on Fridays, 14:15-15:45 hrs, starting on 24/10: INF 227 (KIP), HS 2.

Written exam on Tue., 10/02/15, 10:00-12:00 hrs, INF 227 (KIP), SR 1.404.

Content - Prerequisites - Script - Literature - Additional material - Exercises - Exam

The lecture course provides an introduction to field theoretic methods for systems with many degrees of freedom. A focus will be set on applications to ultracold, mostly bosonic, atomic gases, superfluids and superconductors as they are the subject of many fore-front present-day experiments. Methodologically, the lecture will introduce the basics of the operator as well as the path-integral approach to quantum field theory. In applying these techniques I will in particular concentrate on thermal and dynamical properties of the considered systems. Knowledge of the basics of quantum mechanics and statistical mechanics is presumed while the course is designed to be self-contained on the quantum-field-theory side.

Content:

1. Introduction
   
2. Introduction to quantum field theory of many-body systems
   - The quantum mechanical harmonic oscillator - Classical field theory - Fock space - Coherent states - Free systems and Wick's theorem - Cumulant expansion
   
3. Mean-field theory of a weakly interacting Bose gas
   - Non-linear Schrödinger model - Bogoliubov quasiparticles - Low-energy scattering theory - Interacting ground state - SU(1,1) coherent states - Thermal Bogoliubov quasiparticles
   
4. Path-integral approach to quantum field theory
   - A quick reminder of the Feynman path integral - Functional calculus - Saddle-point expansion and free propagator - Perturbation expansion, Dyson series, and resummation - Correlation functions - Connected functions and cumulants - Feynman diagrammatics - Low-energy effective theory - Linear-response theory - Retarded and advanced Greens functions - Spectral and statistical functions - Thermal path integral - ^*The quantum effective action - ^*Spontaneous Symmetry Breaking
   
5. Low-temperature properties of dilute Bose systems
   - Path-integral representation of the interacting Bose gas - Ginsburg-Landau theory of spontaneous symmetry breaking - The Luttinger-liquid description and the XY model - Superfluid phase transition and spontaneous symmetry breaking - Nambu-Goldstone theorem - Superfluid phase in low dimensions - Finite-temperature superfluids - Dimensionally reduced path integral - Hydrodynamic formulation and vortices - The Berezinskii-Kosterlitz-Thouless transition - Superfluid to Mott insulator transition - ^*Superfluidity and superconductivity - ^*Anderson-Higgs mechanism
   

Prerequisites:

• Quantum Mechanics (PTP 3), Statistical Mechanics (PTP 4); Quantum Field Theory I or Quantum Optics (useful but not a precondition)

Skript :

• Download complete pdf here.
• The Script of the previous lecture in WT 12/13 (with a different focus) can be found here.

Literature:

General texts on quantum field theory

• Brian Hatfield, Quantum Field Theory of Point Particles and Strings. Addison Wesley, Oxford, 2010. [ Google books | HEIDI ]
• Michael E. Peskin, Daniel V. Schroeder An introduction to quantum field theory. Westview, Boulder, 2006. [ Google books | HEIDI ]
• Xiao-Gang Wen, Quantum Field Theory of Many-Body Systems. OUP, Oxford, 2010. [ Google books | HEIDI ]

Quantum optics and phase-space methods

• S.M. Barnett, P.M. Radmore, Methods in Theoretical Quantum Optics, Clarendon Press, Oxford, 1997. [ HEIDI ]
• C.W. Gardiner, Quantum Noise. 2nd Ed. Springer Verlag, Berlin, 2000. [ HEIDI ]
• L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, CUP, Cambridge, 2008 (ISBN 0-521-41711-2). [ HEIDI ]
• W.P. Schleich, Quantum Optics in Phase Space. Wiley-VCH, Weinheim, 2001. [ HEIDI ]

Ultracold atomic gases: General texts and theory reviews

• A. Griffin, D. W. Snoke, S. Stringari (Eds.), Bose-Einstein condensation. CUP, Cambridge, 2002. [ Google books | HEIDI ]
• C.J. Pethick and H. Smith, Bose-Einstein condensation in Dilute Gases. CUP, Cambridge, 2002. [ Google books | HEIDI | Full Text ]
• A. Leggett, Bose-Einstein condensation in the alkali gases: Some fundamental concepts. Review of Modern Physics 73, 307 (2001).
• L.P. Pitaevskii and S. Stringari, Bose-Einstein condensation, Clarendon Press, Oxford, 2003. [ Google books | HEIDI ]
• F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of Bose-Einstein condensation in trapped gases. Review of Modern Physics 71, 463 (1999).
• A. Fetter, Theory of a dilute low-temperature trapped Bose condensate. arXiv.org:cond-mat/9811366 (1998).
• T. Gasenzer, Ultracold gases far from equilibrium. Eur. Phys. Journ. ST 168, 89 (2009); arXiv.org: 0812.0004 [cond-mat.other] .

Ultracold atomic gases: A few original experimental perspectives

• E.A. Cornell, J.R. Ensher, and C.E. Wieman, Experiments in Dilute Atomic Bose-Einstein Condensation. arXiv.org:cond-mat/9903109 (1999).
• W. Ketterle, D.S. Durfee, and D.M. Stamper-Kurn, Making, probing and understanding Bose-Einstein condensates. arXiv.org:cond-mat/9904034 (1999).

Two-body scattering theory

• K. Burnett, P.S. Julienne, P.D. Lett, E. Tiesin, and C.J. Williams, Quantum encounters of the cold kind. Nature (London) 416, 225 (2002).
• J. Dalibard, Collisional dynamics of ultra-cold atomic gases. Proc. Int. School Phys. Enrico Fermi, Course CXL: Bose-Einstein condensation in gases, Varenna 1998, M. Inguscio, S. Stringari, C. Wieman edts.
• C.J. Joachain, Quantum Collision Theory. North-Holland, Amsterdam, 1983. [ HEIDI | Scribd Full Text ]
• L.D. Landau and E. M. Lifshitz, Quantum Mechanics. Non-relativistic theory. (see Chapters XVII & XVIII.) Pergamon Press, Oxford, 1977. [ HEIDI | Online Full Text ]
• R.G. Newton, Scattering Theory of Waves and Particles. Dover publications, 2002. [ HEIDI | Google Books ]

Non-equilibrium quantum field theory and quantum kinetic theory

• L.P. Kadanoff and G. Baym, Quantum statistical mechanics. Addison-Wesley, Redwood City, 1989. [ HEIDI ]
• Jørgen Rammer, Quantum field theory of non-equilibrium states. CUP, Cambridge, 2007. [ Online edition | HEIDI ]
• J. Berges, Introduction to nonequilibrium quantum field theory, AIP Conf. Proc. 739, 3 (2005); arXiv.org: hep-ph/0409233 .
• P. Danielewicz, Quantum Theory of Nonequilibrium Processes, Annals of Physics 152, 239 (1984) .
• M. Bonitz, Quantum kinetic theory. Teubner, Stuttgart, 1998. [ Contents | HEIDI ]
• E. Calzetta and B.-L. Hu, Nonequilibrium quantum field theory. CUP, Cambridge, 2008. [ Online fulltext | HEIDI ]

Additional material

• D. Ebert and V. S. Yarunin, Functional-Integral Approach to the Quantum Dynamics of Nonrelativistic Bose and Fermi Systems in the Coherent-state Representation. Fortschr. Phys. 42, 7, 589 (1994).

Exercises:

Exercises will be held on Fridays, 14:15-15:45 hrs, in HS 2, INF 227 (KIP), starting on 24/10/14. Tutor: Markus Karl. (Please register here.)



Exam:

Passing the written exam, which will prospectively take place on Tue., 10/02/15, 10:00-12:00 hrs, INF 227 (KIP), SR 1.404, will be the condition to obtain 8 CPs for the lecture.